No, it wouldn't. The whole BIS service (at least the push eMail thing) doesn't run on network operators' servers, but on Blackberry's own servers (
that's why Blackberries are considered a security risk). So when you sign up for BIS with your network operator they actually register your device with Blackberry, as operators don't even have the infrastructure for running BIS on their own. And that's where problems start, since Blackberry will refuse to register a device, that exists in their database and is already assigned to an existing BIS account, to a second account.
Just to proof my allegation, check out these BIS registration pages of different network operators, which all run on blackberry.com:
https://bis.eu.blackberry.com/html?brand=tim
https://bis.eu.blackberry.com/html?brand=tre
https://bis.eu.blackberry.com/html?brand=3uk
https://bis.eu.blackberry.com/html?brand=orangeuk
https://bis.eu.blackberry.com/html?brand=vodade
https://bis.eu.blackberry.com/html?brand=o2de
https://bis.eu.blackberry.com/html?brand=eplus
Only if your old operator deletes your BIS account on Blackberry's system, it will be released and you'll be able to re-register for BIS with another operator.
P.S.: Here's one of many sources on the web confirming what I'm explaining:
http://telecominformatics.com/2010/0...e-bis-service/